Given an algebraically closed field, a student wonders about the equivalence of its
multiplicative group and an isomorphic subgroup. Doctor Vogler provides two
counterexamples of injective mappings that are not surjective.

Let G be a finite group, f an automorphism of G such that f^2 is the
identity automorphism of G. Suppose that f(x)=x implies that x=e (the
identity). Prove that G is abelian and f(a)=a^-1 for all a in G.

Can you help me give a description of all Euclidean functions of Z?
The common example is of course the absolute value function, but it
seems to me that other weird Euclidean functions can be constructed, too.